Note: We are going to use the following formula extensively in solving the following problems. The Distance between any two points and is 

Notes: If a point is on , its ordinate is , therefore the point on is taken as . Similarly, if the point is on , its abscissa is , therefore the point on is taken as . For details refer to the following lecture notes.

Question 1: Find the distance between the following pairs of points:

i) and

ii) and

iii) and

iv) and

Answer:

i) and

Distance

ii) and

Distance

iii) and (

Distance

iv)  and

Distance

Question 2: Find the distance between the origin and the points:

i)

ii)

iii)

Answer:

i)

Distance

ii)

Distance

iii)

Distance

Question 3: The distance between point and is . Find .

Answer:

Distance:

Question 4: Find the coordinate of the point on which are at a distance of units from the point .

Answer:

Distance:

Therefore the points are

Question 5: Find the coordinate of the point on which are at a distance of units from the point .

Answer:

Distance:

Hence the points could be

Question 6: A point is at a distance of units from the point . Find the coordinates of the point if its ordinate is twice its abscissa.

Answer:

Distance:

Hence the points could be

Question 7: A point is equidistant from the point and . Find .

Answer:

is equidistant from and

Therefore

Question 8: What point on is equidistant from the point and .

Answer:

Let the point be . Therefore

Therefore the point is

Question 9: What point on is equidistant from the point and .

Answer:

Let the point be . Therefore

Hence the point is

Question 10: A point lies on and another point lies on . Write the ordinate of point , abscissa of point . If the abscissa of point is and ordinate of point is . Calculate the length of the line segment .

Answer:

Let and . Given  and

Distance:

Question 11: Show that the points and are the vertices of an isosceles triangle.

Answer:

and

Therefore two sides are equal which makes it an isosceles triangle.

Question 12: Prove that the points and are the vertices of the rectangle .

Answer:

and

Therefore and .

Hence it is a rectangle.

Question 13: Prove that the points and are the vertices of an isosceles triangle. Find the area of the triangle.

Answer:

and

For this to be a right angled triangle we should have

. Hence proved that it is a right angled triangle.

Area sq. units.

Question 14: Show that the points and are the vertices of the square .

Answer:

and

Therefore .

Hence it is a square.

Question 15: Show that and are the vertices of a rhombus.

Answer:

and

Therefore .

Two sides are equal and the other two sides are of different length. Hence it is a rhombus.

Question 16: Points and are the vertices of a quadrilateral . Find a if a is negative and .

Answer:

and

Therefore 

. Hence  as it is negative.

Question 17: The vertices of a triangle are and . Find the coordinates of the circumcenter of the triangle.

Answer:

and are the points

Let the coordinates of the circumcenter

Therefore  

Hence  

Therefore equation 1:

Also equation 2:

Hence the coordinates of the circumcenter is

Question 18: Given and . Find if .

Answer:

and

Therefore

Question 19: Given and . Find if .

Answer:

and

Question 20: The center of the circle is . Find if the circle passes through and the length of the diameter is units.

Answer:

Diameter  units i.e.Radius  units

Question 21: The length of the line is units and the coordinates of are , calculate the coordinates of point , if its abscissca is .

Answer:

and Let